To provide small insight into the work of people who actually do doubt ZFC I'd like to source two interesting books. First is Peter Aczel's work with the Anti-

To provide small insight into the work of people who actually do doubt ZFC I'd like to source two interesting books. First is Peter Aczel's work with the Anti-foundation Axiom which allows you to create sets with "infinite regress". Instead of blanket disallowing these kinds of set like ZFC does, he enlarges ZFC by allowing any set which is a unique solution to a group of equations that may be self-referential. I was introduced to this stuff by a book "Vicious Circles" by Barwise and Moss (http://www.amazon.com/Vicious-Circles-Center-Language-Information/dp/1575860082 http://www.amazon.com/Vicious-Circles-Center-Language-Inform...). The important point is that the Anti-Foundation Axiom lets us model infinite, streaming structures similar to those modeled in computer programs. It also motivates the somewhat unpopular idea of bisimulation which is very necessary for creating a kind of equality in an AFA world... and also for proving the equality of streaming algorithms. Second is Lawvere's introduction to thinking about the foundations of mathematics from a Category Theoretic perspective. Lawvere is a proponent of thinking of Set theory as simply one, somewhat interesting Category which can be generated by a more foundational theory and set of axioms. I'm not personally anywhere nearly well-knowing enough to say whether that works, but his book Conceptual Mathematics (http://fef.ogu.edu.tr/matbil/eilgaz/kategori.pdf http://fef.ogu.edu.tr/matbil/eilgaz/kategori.pdf) gives a very interesting POV on how to work from the Category Theoretical basis to answer some normal Set theoretic questions.